Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19651 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C23×C4 |
| direct product | C53×#19659 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C53×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(17,18,24,19)(20,22,26,23)(21,25,32,27)(28,30,29,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(2,6)(7,14)(8,13)(9,15)(17,19,22,26,24,18,23,20)(21,27,30,29,32,25,31,28);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(17,20,23,18,24,26,22,19)(21,29,31,27,32,28,30,25);(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 12647);